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CARABALLO, SIMON
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Thermomechanical beam element for analyzing stresses in functionally graded materials
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by SIMON CARABALLO.
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[Tampa, Fla] :
b University of South Florida,
2011.
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Title from PDF of title page.
Document formatted into pages; contains 120 pages.
Includes vita.
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Disseration
(Ph.D.)University of South Florida, 2011.
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Includes bibliographical references.
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Text (Electronic dissertation) in PDF format.
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ABSTRACT: Modeling at the structural scale most often requires the use of beam and shell elements. This simplification reduces modeling complexity and computation requirements but sacrifices the accuracy of throughthethickness information. Several studies have reported various design approaches for analyzing functionally graded material structures. One of these studies proposed a twonode beam element for functionally graded materials (FGMs) based on first order shear deformable (FOSD) theory. The derivation of governing equations included spatial temperature variation. However, only the constant temperature case was carried through in the element formulation. This investigation explore the effects of spatial temperature variation in the axial and throughthethickness direction of this proposed element and present a new standard threenode beam finite element modified for structure constructed of FGMs. Also, the influence of the temperature dependency of the thermoelastic material properties on the thermal stresses distribution was studied. In addition, variations in the layer thicknesses within multilayer beam models were studied to determine the effect on stresses and factor of safety. Finally, based on the specific factor of safety, which combines together the strength and mass of the beam, the best layer thicknesses for the beam models were established. The key contributions expected from this research are: 1. development and implementation of a threenode beam element as a finite element code into the commercial computational tool MATLAB to analyze thermomechanical stresses in structures constructed of functionally graded materials; 2. a strategy to simulate different load cases in structures constructed of functionally graded materials; 3. an analysis of the influence of the FGM interlayer thickness on the factor of safety/specific gravity ratio in structures constructed of functionally graded materials under thermomechanical loads; 4. and an analysis/comparison of the advantages/benefits of using structures constructed of functionally graded materials with respect to those constructed with homogenous materials.
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Mode of access: World Wide Web.
System requirements: World Wide Web browser and PDF reader.
590
Advisor:
Kaw, Autar K.
653
Finite Element Formulation
Multilayer Beams
Nonhomogeneous Material Properties
Numerical Simulation
Temperaturedependent Properties
Thermal Stresses
690
Dissertations, Academic
z USF
x Engineering Mechanical Engineering
Doctoral.
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t USF Electronic Theses and Dissertations.
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u http://digital.lib.usf.edu/?e14.4865
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Thermo M echanical B eam E lement for A nalyzing S tresses in F unctionally G raded M aterials by Simn A. Caraballo A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mechanical Engineering College of Engineering University of South Florida Major Professor: Autar K. Kaw Ph.D Kandethody M. Ramachandran, Ph.D. Muhammad M. Rahman, Ph.D. Glen H. Besterfield, Ph.D. Ali Yalcin, Ph.D. Date of Approval: M arch 29 20 11 Keywords: M ultilayer B eams T hermal S tress es, N onhomogeneous M aterial P roperties, T emperature D ependent P roperties, F inite E lement F ormulation N umerical S imulation Copyright 20 11 Simn A. Caraballo
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Dedication First, I give praise and honor to my Lord and savior Jesus Christ for allowing me to accomplish this goal. I dedicate t his dissertation to my beloved wife Marisolin and my daughter Marielsy ; without their love and emotional support this work would have not been possible. I would like to thank them for their sacrifice, patience and love, and for all that they have given to me in my life. I would like to express my sincere thanks to my advisor, Professor Autar K. Kaw for his constructive guidance, financial support, understanding and patience throughout the study, especially when I was in down moments. My appreciation extends as well to the members of my supervisory committee ; Dr. Muha mmad M. Rahman and Dr. Glen H. Besterfield from the Department of Mechanical Engineering, Dr. Kandethody M. Ramachandran from the Department of Mathematics, and Dr. Ali Yalcin from the Department of Industrial and Management Systems I also wou ld like to e xpress my sincere thanks to Ms. Marsha Brett Ms. Kate Johnson and Dr. Rajiv Dubey from the College of Engineering for their financial support. I wish to express my sincere gratitude to my parents for their unconditional love and confidence in me all the time. Last but no t least the financial support of the Experimental Polytechnic National
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i Table of Contents List of Tables ................................ ................................ ................................ ..................... iii List of Figures ................................ ................................ ................................ .................... iv Nomenclature ................................ ................................ ................................ .................... vii Abstract ................................ ................................ ................................ .............................. ix Chapter 1 Introduction ................................ ................................ ................................ ........ 1 Motivation ................................ ................................ ................................ ....................... 1 Research Goals ................................ ................................ ................................ ................ 2 Outcomes ................................ ................................ ................................ ........................ 3 Dissertation Organization ................................ ................................ ............................... 4 Chapter 2 Review of Relevant Literature ................................ ................................ ........... 6 Introduction ................................ ................................ ................................ ..................... 6 Chapter 3 Theoretica l Background ................................ ................................ ..................... 9 Introduction ................................ ................................ ................................ ..................... 9 FGM Theoretical Fundamentals ................................ ................................ ..................... 9 Conceptual Idea of FGMs ................................ ................................ ........................... 9 Effective Properties of FGMs ................................ ................................ ................... 14 Consideration of Tempe rature Dependence of Material Properties for FGMs ......... 19 FGMs Applications ................................ ................................ ................................ ... 30 Chapter 4 Formulation of Governing Equations ................................ ............................... 32 Introduction ................................ ................................ ................................ ................... 32 Beam Theory for FGM Structures ................................ ................................ ................ 32 Finite Element Formulations ................................ ................................ ......................... 36 Two node Element Formulation ................................ ................................ ............... 36 Thre e node FOSD Element ................................ ................................ ....................... 43 Temperature Profile Modeling ................................ ................................ ...................... 48 One dimensional Heat Conduction Steady State Exact Solution for a 3 layer FG Beam ................................ ................................ ................................ ......................... 48 Two dimensional Heat Conduction Steady State Numerical Solution for a 3 Layer FG Beam with Temperature Dependency of the Material Properties ......... 53 Chapter 5 Analyses and Results ................................ ................................ ........................ 57
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ii Introduction ................................ ................................ ................................ ................... 57 Comparisons of the Element Formulation Simulations with Related Literature .......... 57 ................................ .................... 58 Comparison with Chakraborty et al. Models ................................ ............................ 62 Simulations with Generic Temperature Distr ibutions and Temperature Independence of the Material Properties ................................ ................................ .............................. 68 Simulations with Actual Temperature Distributions with and without Temperature Dependence of the Material Properties ................................ ................................ ......... 77 Influence of the Interlayer Thickness on the Factor of Safety in Compo site Beams .... 92 Determination of the Baseline Thickness of the Metallic Layer for Studying the Influence of the FGM Interlayer in the Factor of Safety ................................ .............. 93 Effect of Thickness of the Graded Interlayer in the Factor of Safety for the Tri layer Model ................................ ................................ ................................ ............................ 95 Chapter 6 Conclusions and Future Work ................................ ................................ ........ 100 Introduction ................................ ................................ ................................ ................. 100 Conclusions ................................ ................................ ................................ ................. 100 Recommendations and Future Work ................................ ................................ ........... 103 References ................................ ................................ ................................ ....................... 104 About the Author ................................ ................................ ................................ ............ End
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iii List of Tables Table 1. Effective property formulas of FGMs [24]. ................................ ....................... 18 Table 2. Thermal properties of steel [31, 35, 3 9]. ................................ ............................ 26 Table 3. Thermal properties of alumina [31, 38]. ................................ ............................ 26 Table 4. Material and geometrical parameters of a tri layered beam .............................. 51 Table 5. Thermo elastic properties of nickel and alumina at 300 K ................................ 60 Table 6. Thermo elastic properties of steel and alumina at 300 K ................................ .. 63 Table 7. Loading cases for models from literature paper [1]. ................................ .......... 63 Table 8. Temperature distributions. ................................ ................................ ................. 71 Table 9. Factor of safety for the different models. ................................ ........................... 91 Table 10. Layer thickness variation for the bi material model. ................................ ........ 93 Table 11. Layer thickness variation for the 3 layer model. ................................ .............. 96
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iv List of Figures Figure 1. Examples of material grading in functionally graded materials. ..................... 10 Figure 2. Graphical FGM representation of gradual transition in the direction of the temperature gradient. ................................ ................................ ...................... 12 Figure 3. Ceramic volume fraction throughout the FGM layer ................................ ...... 14 Figure 4. Effect of the grading parameter n on the volume fraction V c .......................... 15 Figure 5. Material properties throughout the FGM layer. ................................ ............... 16 Figure 6. Temperature dependence of elastic modulus and thermal conductivity for aluminum titanate ceramics. ................................ ................................ ........... 20 Figure 7. Temperature dependence of ther mal conductivity for several engineering materials. ................................ ................................ ................................ ......... 21 Figure 8. Temperature dependence of the linear thermal expansion for sever al engineering materials. ................................ ................................ ..................... 2 3 Figure 9. materials. ................................ ................................ ................................ ......... 24 Figure 10. Temperature dependence of the flexural strength for several engineering materials. ................................ ................................ ................................ ......... 25 Figure 11. Temperature dependence of the thermoelastic properties of steel. ................. 28 Figure 12. Temperature dependence of the thermoelastic properties of alumina. ............ 29 Figure 1 3. FGM application for a turbine blade design. ................................ ................... 31 Figure 14. FGM application for relaxation of stress concentration in lathe bits. ............. 31 Figure 15. Beam coordinate system. ................................ ................................ ................. 34 Figure 16. Nodes and degrees of freedom for the 2 node element ................................ ... 40
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v Figure 17. Nodes and degrees of freedom for the 3 node element ................................ ... 44 Figure 18. Three layer beam with perfect thermal contact at the interface. ..................... 49 Figure 19. Depth wise exact temperature distribution obtained from the solution of the heat conduction differential equation. ................................ ....................... 52 Figure 20. Three layer beam geometry and boundary conditions ................................ .... 53 Figure 21. Geometry and nomenclature for a tri layered composed beam model from literature reference [1]. ................................ ................................ .................... 59 Figure 22. Axial thermal stress distribution in a Ni Graded Layer Al 2 O 3 trilayer beam subject to a T = 100 o C. ................................ ................................ ............... 61 Figure 23. Geometry and loading cases for models from literature paper [1]. ................. 63 Figure 24. Axial stress through the thickness for case 1. ................................ .................. 65 Figure 25. Transverse shear stress through the thickness for case 1. ................................ 65 Figure 26. Axial stress through the thickness for case 2. ................................ .................. 66 Figure 27. Axial stress through th e thickness for case 3. ................................ .................. 66 Figure 28. Transverse shear stress through the thickness for case 3. ................................ 67 Figure 29. Beam configurations. ................................ ................................ ....................... 68 Figure 30. Beam geometry and boundary conditions. ................................ ...................... 70 Figure 31. Normalized axial stress through the thickness for case 1, T =100. ................ 72 Figure 32. Normalized axial stress through the thickness for case 2, T ( z ) = ................................ ................................ .................. 73 Figure 33. Normalized ax ial stress through the thickness for case 3, T ( x ) = ...... 74 Figure 34. Normalized transverse shear stress through the thickness for case 3, T ( x ) = ................................ ................................ ................................ 74 Figure 35. Normalized axial stress through the thickness for case 4, T ( x,z ) = ................................ ................................ ........ 76 Figure 36. Normalized transverse shear stress through the thickness for case 4, ................................ ...................... 76
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vi Figure 37. Beam geometry and boundary conditions (Bimaterial) ................................ ... 78 Figure 38. Thermal conductivity distribution with and without temperature dependence (Bimaterial case). ................................ ................................ ........ 79 Figure 39. Temperature profile with and without temperature dependence (Bimaterial case). ................................ ................................ ............................ 79 Figure 40. Beam geometry and boundary conditions (Average interlayer) ...................... 81 Figure 41. Thermal conductivity distribution with and without temperature dependence (Average interlayer case). ................................ ........................... 82 Figure 42. Temperature profile with and without temperature dependence (Average interlayer case). ................................ ................................ ............................... 82 Figure 43. Beam geometry and boundary conditions (FGM interlayer) ........................... 83 Figure 44. Thermal conductivity distribution with and without temperature dependence (FGM interlayer case). ................................ ................................ 85 Figure 45. Temperature profile with and without temperature dependence (FGM interlayer case). ................................ ................................ ............................... 85 Figure 46. Normalized axial stress through the thickness for actual temperature distribution. ................................ ................................ ................................ ..... 87 Figure 47. Normalized transverse shear stress through the thickness for actual temperature distribution. ................................ ................................ ................. 89 Figure 48. Factor of safety for the different models ................................ ......................... 90 Figu re 49. Beam models for studying the effect of the FGM interlayer thickness in the factor of safety. ................................ ................................ .......................... 92 Figure 50. Effect of t hickness of FGM interlayer in the factor of safety for the tri layer model ................................ ................................ ................................ 97 Figure 51. Effect of thickness of FGM interlayer in the specific factor of safety for the tri layer model ................................ ................................ ........................... 98
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vii Nomenclature The following symbols are used in this dissertation: stiffness coefficients thermal stiffness coefficients beam length shear modulus element length strain energy beam thickness axial displacement transverse displacement axial s train coefficient of thermal expansion rotation of reference axis a bout y axis shear strain axial s tress
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viii shear stress temperature change from reference state mechanical force vector first thermal force vector second thermal force vector element displacement vector nodal displacement vector elem ent thermal vector nodal thermal vector element stiffness matrix strain displacement matrix s hape function Subscripts b value at bottom of beam t value at top of beam Superscripts with respect to reference plane
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ix A bstract Modeling at the structural scale most often requires the use of beam and shell elements. This simplification reduces modeling complexity and computation requirements but sacrifices the accuracy of through the thickness information. Several studies have rep orted various design approaches for analyzing f unctionally g raded material structures. One of th ese studies proposed a two node beam element for f unctionally g raded m aterials ( FGMs ) based on first order shear deformable ( FOSD ) theory. The derivation of governing equations included spatial temperature variation. However, only the constant temperature case was carried th r ough in the element formulation. This investigation explore the effects of spatial temperature variation in th e axial and through the thickness direction of th is proposed element and present a new standard three node beam finite element modified for structure constructed of FGMs. Also, t he influence of the temperature dependency of the thermo elastic material properties on the thermal stresses distribution was stu died. I n addition variations in t he layer thicknesses within multilayer beam models w ere studied to determine the effect on stresses and factor of safety Finally, based on the specific factor of safe ty, which combines together the strength and mass of the beam, the best layer thicknesses for the beam models were established.
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x The key contributions expected from this research are: 1. development and implementation of a three node beam element as a finite element code in to the commercial computational tool MATLAB to analyze thermo mechanical stresses in structures constructed of functionally graded materials ; 2. a strategy to simulate different load cases in structures constructed of functionally grade d materials ; 3. a n analysis of the influence of the FGM interlayer thickness on the f actor of s afety/ specific gravity ratio in structures constructed of functionally graded materials under thermo mechanical loads ; 4. and an analysis/comparison of the advantages/benefits of using structures constructed of functionally graded materials with respect to those constructed with homogenous materials
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1 Chapter 1 Introduction Motivation The main benefit of using f unctionally g raded m aterials (FGMs) instead of traditional materials is that the int ernal composition of their component materials can be tailored to satisfy the requirements of a given structure Although much of this technology has not been fu lly implemented the internal structure of the material could be prepared to manufacture hybrid high temperature pressure vessels or other thermal structures. B efore attempting to fabricate complicated applications out of FGMs, it is very important that th e tools for structural analysis are developed This work is an important step in being able to properly design mechanical structures using a functionally graded material system. On a grander scale, the ultimate goal of this research is to help determine if the structures constructed of functionally graded materials can be used instead of traditional materials within the context of the needed applications. One of these applications is a space shuttle, where the aluminum substructure is shielded by a thermal protection system (TPS) barrier consisting of several layers of primers, tile, adhesives, fibers, and coatings. The core metallic of structures made of FGM s could resist higher temperatures and the structure size requirements can be reduce d Additionally mass could be
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2 minimized by tailoring the ingredient of each component based upon the load and stress interactions present in different areas of the mechanical structure Functionally graded materials (FGM s ) are of increasing importance as designers seek a way to address structures under combined thermal and mechanical loads. The finite element method is commonly employed to analyze structures where beam, plate/shell, and solid elements are used. The question arises as to how to implement element formulation s for structures composed of FGMs. As a n important step to achieve this goal a first order shear deformable (FOSD) beam model is investigated and applied to beams subjected to spatial variations in temperature. Research Goals T he goal of this research i s to determine if the structures constructed of functionally graded materials can be used instead of traditional materials The study will focus on the modeling and simulation of : 1. Functionally graded beam structures with material properties varying throughout the thickness of the beam 2. Functionally graded beam structures with temperature dependen t thermo elastic material properties. 3. Elastic therm o mechanical stresses in FGM structures
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3 Outcome s The major outcomes of the present research are the following : 1. Develop a finite element program to analyze thermo mechanical stresses in structures constructed of functionally graded materials. 2. The p erformance of the proposed element formulation is presented throughout comparisons with FGMs model avail able in related literature. 3. Methodology to conduct analytical and numerical simulations of thermal loading studies conducted on the FGMs beam structures in one and two dimension s 4. Simulate structures constructed of functionally graded materials with and without t emperature d ependence of the m aterial p ropertie s 5. Analyze the influence of layer thicknesses within multilayer beam models on the f actor of s afety/ specific gravity ratio in structures constructed of functionally graded materials under thermo mech anical loads. 6. Analyze and compare the advantages and benefits of using structures constructed of functionally graded materials with respect to those constructed with homogenous materials.
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4 Dissertation Organization This research work is organized where Chapter 1 presents an introduction motivation, the research goals and major outcomes of this work. Then Chapter 2 discuss es previous research work on the analysis and design of FGM structures. A review of current research in this area is also introduced. Chapter 3 presents fundamental theoretical aspects of FGMs and their applications. It also introduces t he c onceptual idea of FGMs and their distinct features in comparison with other engineering material s Approaches for modeling and calculating the effective properties of FGMs are discussed. Considerations of the t emperature dependence of material properties for FGMs are presented. Typical engineering applications of FGMs are also provided Chapter 4 presents a detailed formulati on of the g overning e quations for analyzing functionally graded material models. Formulations of the equations of motion are developed for a first order s hear deformable beam Two finite element formulations are presented. The first formulation is a two node formulation based on Chakraborty et al [ 1 ] where a beam element is developed to study the thermo elastic behavior of functionally graded beam structures This is followed by a new three node element formulation Chapter 5 discusses the results of the analyses performed in this research Numerical s imulations of t hermal l oading studies conducted on the FGMs beam structures are presented Also, this chapter describes a study to determine the influence of
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5 manipulating the FGM layer thickness on the f actor of s afety and the specific factor of safety in these structures under thermal loads Finally, Chapter 6 presents a summary of the work conducted and a discussion of the main conclusions drawn. The chapter also offers recommendations that emerge from this work for future research in the field.
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6 Chapter 2 Review of Relevant Literature Introduction This chapter summarizes previous research work in the analysis and design of FGM structures. The extensive research in this field, which started with the pioneering work of Suresh an d Mortensen [ 2 ] Reddy [ 3 ] and Sankar [ 4 ] has led to the development of several design approaches for analyzing FGM structures that are currently used in many applicati ons throughout the world. A brief description of the most applicable works in this area is presented as follows Reddy [ 5 ] worked on characterizing the theoretical formulation o f FGMs to include the derivations of equations used to calculate material properties throughout the thickness of the material based on the through the thickness distribution of materials. Na and Kim [ 6 ] s tudied the thermo mechanical buckling of FGMs using a finite element discretization method. Cooley [ 7 ] researched FGM shell panels under thermal loading also using the finite element method. Hill and Lin [ 8 ] concentrated their research of FGMs in the field of residual stress measurement in a ceramic metallic graded material using experimental procedures that released residual stresses by ma king incisions into the material and measuring the resulting change in stress with strain gages. Hill et al [ 9 ] participated in studying the fracture testing of layered (as opposed to a continuous function) FGMs.
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7 Some work in the area of FGM aluminum matrix composites include the study of Kang and Rohatgi [ 10 ] who performed a transient thermal analysis of solidification in a centrifugal casting for composite materials containing pa rticle segregation. Another contributor to the field, Sankar [ 4 ] showed that a functionally graded Euler Bernoulli beam is subject to the same limitations normally associated with beam theory under mechanical loading. For comparison between a beam theory and the elasticity solution a simply supported beam with a si nusoidal distributed load was solved. Later Sankar and Tzeng [ 11 ] [ 4 ] earlier work by investigating beams with through the thic kness temperature gradients. Additionally, Chakraborty et al [ 1 ] proposed a two node beam element for FGMs based on FOS D theory and applied it to static, thermal, free vibration and wave propagation problems. The assumed displacement field of the element satisfies the general solution to the static part of the governing equations. Static condensation ( Cook et al. [ 12 ] Wilson [ 13 ] ) is used to reduce the number of unknowns in the elements displacement field to the number of degrees of freedom within the element. T he derivation of governing equations included spatial temperature variation Even though Chakraborty et al [ 1 ] work constitutes an important contribution to t he FGMs field it present s some limitation s. For example, only the one dimensional constant temperature case was carried th r ough in the finite element formulation. Another limitation is that the temperature dependency of the material properties was not considered for the analyzed models. To address these limitations, t his work will investigate the effects of spatial two dimensional temperature variation in the axial and through the th ickness direction of the
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8 element proposed by Chakraborty et al [ 1 ] Additionally, a more accura te three node beam element will be formulated for analyzing the FGMs structures Much more important, since FGM s structures are usually subjected to high ranges of temperatures ( 20 C 800 C ) temperature dependen cy of the material properties is considered in this investigation which will produce more realistic simulations and analys e s of the structures being studied
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9 Chapter 3 Theoretical Background Introduction This chapter presents the fundamental theoretical aspects of FGMs and their applications First, t he c onceptual i dea of FGMs and their distinct features in comparison with other engineering material s is introduced Also, approaches for modeling and calculating the effective properties of FGMs are discussed. In addition, important considerations of the t emperature dependence for FGM material properties are presented. Finally some typical engineering a pplications of FGMs are reviewed FGM Theoretical F undament al s Conceptual I dea of FGMs The term functionally graded materials (FGM s ) refers to solid object s or parts that usually consist of multiple materials or embedded components that is they are materially heterogeneous. The term multiple material objects with clear material domains [ 14 ] A FGM consists of a material whose properties change from one surface to another according to a smooth continuous function based on the position throughout the
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10 thickness of the material. Most often this material consists of ceramic and metallic constituents. One surface is generally a pure metal while the opposite surface is usually pure ceramic or a majority ceramic. The metal portion of the material acts in the role of a structural support while the cer amic provide s thermal protection when subjected to harsh temperatures. The function describing the material variation throughout the material and more importantly the material property variation makes it possible to tailor the function to suit the needs of various applications. Examples of different types of material grading in functionally graded materials are shown in Figure 1 as presented by Refs. [ 1 5 17 ] a) Examples of grading sources [ 15 ] b) Planar grading [ 16 ] c) Example of localized material grading [ 17 ] Figure 1 Examples of material grading in functionally graded materials Reprinted from Refs. [ 15 17 ] with permission from Elsevier
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11 The continuous change in the microstructure of FGMs distinguishes them from the fiber reinforced laminated composite materials, which have a mismatch of mechanical properties across an interface due to two discrete materials bonded together. As a result the constituents of the fiber matrix composites are prone to debond ing at extremely high thermal loading. Also, the anisotropic constitution of laminated composite structures often results in stress concentrations near material and geometric interfaces that can lead to damage in the form of delamination, matrix cracking, and adhesive bond separation [ 5 ] C ontinuous or nearly continuous gradual change in material properties of FGMs reduces significantly these problems, making them a desirable cho ice for adverse thermal gradient applications. FGMs alleviate these problems because they consist of a continuous variation of material properties from one surface to the other. The continuous nature of the variation lessens the stress concentrations whic h become troublesome in a laminated composite material. Also the smooth transition through the various material properties reduces both thermal and residual stresses [ 18 ] In most cases the material progresses from a metal on one surface to a ceramic or mostly ceramic on the opposite surface, with a s mooth transition throughout the center of the material. Also the material properties can change in any orientation across a material, but the majority of applications to date deal with a material in which the properties change through the thickness of the material. The material transitions from a metal to a ceramic by increasing the percentage of ceramic material present in the metal until the appropriate percentage is reached or a pure ceramic is achieved (See Figure 2 ).
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12 Figure 2 Graphical FGM representation of gradual transition in the direction of the temperature gradient. Reprinted from Ref. [ 19 ] with p ermission from Elsevier. Since the material does not have a dramatic change in material properties at any one point through the thickness, it would not cause a large stress concentration. This material usually exists where there is an extreme temperature gradient which is designated by T hot and T cold in Figure 2 The ceramic face of the material is generally exposed to a high temperature, while the metallic face is us ually subjected to a relatively cooler temperature. The smooth transition of material properties allows for a material whose properties provide thermal protection as well as structural integrity reducing the possibilities of failure within the structure. T his reduction of failure is of critical importance in space programs where thermal protection tiles are laminated to the metallic structure of the space shuttle to handle the extreme temperatures during re entry into the susceptible to cracking and debonding at the superstructure/tile interface due to abrupt transition between thermal expansion coefficients. The smooth transitions between material properties reduces the potential cracking and debonding of thermal protecti on tiles laminated to structural members.
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13 The capabilities of the FGMs are quite flexible as one can vary the materials used as well as the function of composition throughout the material at which they transition from surface to surface. A specific metal and ceramic can be chosen for the particular application to capitalize on the positive characteristics of each of the materials. Also, the function between the two outside materials can be mathematically maximized and tailored specifically to meet the need s of the desired application as shown by [ 20 ] Functionally graded materials (FGMs) are new advanced multifunctional composites where the volume fractions of the reinforcements phase(s) (or dispersoids) vary smoothly. This is achieved by using reinforcements with different properties, sizes, and shapes as well as by interchanging the functions of the reinforcement and matrix phases in a continuous manner. The result is a microstructure bearing continuous changes in thermal and mechanical properties at the macroscopic or continuum scale [ 21 ] In other words, FGM is usually a combination of two materials o r phases that show a gradual transition of properties from one side of sample to the other. This gradual transition allows the creation of superior and multiple properties without any mechanically weak interface. Moreover, the gradual change of properties can be tailored to different applications and service environments. This new concept of materials engineering hinges on materials science and mechanics due to the integration of the material and structural considerations into the final design of structural components. Because of the many variables that control the design of functionally graded microstructures, full utilization of the FGMs potential requires the development of appropriate modeling strategies for their response to combined thermo mechanical l oads.
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14 FGMs are ideal candidates for applications involving severe thermal gradients, varying from thermal structures in advanced aircraft and aerospace engines to computer circuit boards. These materials were introduced to take advantage of ideal behavior of its constituents, for example, heat and corrosion resistance of ceramics together with mechanical strength and toughness of metals [ 22 ] Effective Properties of FGMs T o study FGMs a model must be created that describes the function of composition throughout the material. In Figure 3 t he volume fraction , describes the volume of ceramic at any point throughout the thickness according to a pa rameter n which controls the shape of the function ( as see n in Figure 4 ). is given by ( 1 ) Figure 3 Ceramic v olume fraction throughout the FGM layer 0 Graded Layer x z
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15 It follows that the volume fraction of metal , in the FGM is A graphical representation of various values of the parameter n can be seen in Figure 4 Figure 4 Effect of the grading p arameter n on the volume fraction V c The area to the right of each line represents the amount of metal and the area to the left represents the amount of ceramic in t he material. It should be noted that as the material approaches to a hom ogeneous ceramic, while as the material becomes entirely metal. For the material will contain both metal and ceramic. When the distribution is linear containing equal portions of ceramic and metal. According to Nakamura and Sampath [ 23 ] the values of n should be taken in the range of [ 1 / 3 3 ], as value s outside this range will produce an FGM having too mu ch of one phase
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16 One of the most common methods to determine the effective properties of FGMs is the rule of mixtures where t he material properties through the thickness vary as a function of the volume fraction and are given by ( 2 ) As Figure 5 shows, t he variables and are the material property at the top and bottom, respectively c orresponds to the mat erial property of the pure ceramic and P b corresponds to the material property of the pure metal. is given by Eq. ( 1 ) Figure 5 Material properties throughout the FGM layer Even though the rule of mixture s given by E q ( 2 ) is widely used for most researchers in the FGM field t his rule is very general and it does not always give a realistic value of the properties in qu estion. I n fact, mo re appropriate formulas ha ve been found by Nemat Alla [ 24 ] to address the limitations of the rule of mixtures 0 Graded Layer x z
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17 For the analysis of the FGM in this research, formulas show n in Table 1 will be used for estimating the effective values of the thermo mechanical properties. It should be noted that these formula s are particular cases of zero material porosity [ 24 ] At this point, it is important to mention that in th e formulas given in Table 1 the thermo mechanical properties of each material in the composite beam are also a function of the temperature. Th e influence of the temperature on the material properties will be discussed in detail in he T emperature D ependence of M aterial P roperties for FGM section in this chapter.
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18 Table 1 Effective property formulas of FGMs [ 24 ] Mate rial property Effective property formula T hermal conductivity ( ) M odulus of elasticity ( ) Coeff icient of thermal expansion ( ) Poisson r atio ( ) D ensity ( ) Y ield strength ( ) In Table 1 and are the bulks modulus and modulus of rigidity respectively Also, the undefined parame ters are given by , The subscripts and stand for the material property at the top and bottom, respectively for the corresponding property. c orresponds to the material property of the pure ceramic and corresponds to the material property of the pure metal
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19 Consideration of Temperature D ependen ce of Material Properties for FGMs FGMs are generally used in application s where high temperature environments/ field s are involved. In these high temperature environments, some material properties ( t hermal conductivity ( ), coefficient of thermal expansion ( ), m odulus of elasticity ( ) and y ield strength ( ) are of particular pertinen ce to this work) become temperature dependent [ 25 ] In fact the composite beam model structures that will be analyzed in Chapter 5 are subject to high levels of temperature. Therefore, this section review s important aspect s of the influence of tempera ture in the thermo mechanical properties of the materials used in the composite model s to be studied in this work. The influence of the temperature on the material properties have been reported by various researchers and in handbooks of engineering materials For example, Chen and Awaji [ 26 ] stud ied the temperature dependence of the mechanical properties of aluminum titanate (AT) and found that both the fracture strength and fracture toughne ss increased considerably with increase in temperature. They also found the temperature dependence of elastic modulus and thermal conductivity of AT ceramics as show n in Figure 6 Also, Yang et al. [ 27 ] present ed thermo mec hanical post buckling analysis of cylindrical panels that are made of FGMs with temperature dependent thermo elastic properties. They found that the temperature independent solutions are about 9 18% higher than the temperature dependent solutions, that is, the buckling temperature is considerably overestimated when the temperature dependence of the material properties is not taken into consideration. Finally, Richerson [ 28 ] and Murray [ 29 ] present several engineering materials frequently used in high temperature applications and how their thermo elastic properties vary with temperature (see Figure 7 10 )
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20 Figure 6 Temperature dependence of elastic modulus and thermal conductivity for aluminum titanate ceramics Reprinted from Ref. [ 26 ] with permission from Els evier Figure 6 shows the thermal conductivity temperature dependence for several engineering materials. Clearly, we can see that the temperature has a strong effect on the thermal conductivity of ceramics materials. While in most ceramic materials k decreases as T increases, in other materials k increases with T Platinum ha s high thermal conductivit y that increase s with temperature up to at least 120 0 It can be observed that the materials with complex crystal structures have lower thermal conduct ivities. Also the presence of foreign atoms decreases the thermal conductivity [ 30 ] For example, zirconia stabilized with MgO or CaO has low thermal conductivity a nd is very useful as a high temperature refractory material. The highest conductivities are achieved in the least complex structures, that is structures consisting of a sin gle element or similar atomic weight or with no extraneous atoms in solid solution. When comparing to metals, in general ceramics (nonconductor materials) have lower thermal conductivities than metals (conductor materials). However, nonconductor materials such as beryllium oxide,
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21 diamond, and silicon carbide are exceptions to this rule [ 31 ] A detailed discussion on thermal conductivity is beyond the scope of this work; readers interested in a more detailed treatment of thermal conductivity are referred to the literature in Ref. [ 32 ] Figure 7 Temperature dependence of thermal conductivity for several engineering materials. Reprinted from Ref. [ 32 ] with permission from Taylor and Francis Group, LLC. Figure 8 shows the linear thermal expansion as a function of temperature for several materials. Between room temperature and 400 hermal expansion is relatively small for mullite and alumina compare d to polyethylene, nylon, and aluminum alloys. Above 400 this trend is reverted. In general for temperatures above 400 zircon,
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22 mullite, alumina, ZrO 2 and Ni base superalloy exhibit the greatest expansion while fused SiO 2 and aluminum silicate (LiAlSi 2 O 6 ) have the least thermal expansion. It is convenient to mention that the last two materials (SiO 2 and LiAlSi 2 O 6 ) have very little dimensional change as a function of temperature an d can therefore withstand extreme thermal cycling or thermal shock witho ut fracturing [ 33 ] Low thermal expansion ceramic materials have broad potential for both domestic and industrial applications. Fused silica is one of the best t hermal shock resistant materials available. One of the best application example of th is material is put in practice when it is fabricated in a porous foam which is used for lining critical surface of the space shuttle that are exposed to high temperature during ascent and reentry to the atmosphere. This material combine s t he low thermal expansion to prevent thermal shock damage and the very low thermal conductivity to protect the underlying structures which are less thermal resistant [ 33 ]
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23 Figure 8 Temperature dependence of the linear thermal expansion for several engineering materials Reprinted from Ref. [ 32 ] with permission from Taylor and Francis Group, LLC
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24 Figure 9 Temperature dependence of materials Reprinted from Ref. [ 32 ] with permission from Taylor and Francis Group, LLC Figure 9 depicts the effect of the temperature on the elastic modulus of typical ceramics. As we can observe, for each material E decreases slightly as the temperature increases. SiC and TiC have the highest moduli, followed by Al 2 O 3 Si 3 N 4 MgO, and ThO 2 ZrO 2 and MgAl 2 O 4 have relatively low moduli, and LiAlSi 2 O 6 ha s the lowest modul us of this group.
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25 Figure 10 Temperature dependence of the flexural strength for several engineering materials Reprinted from Ref. [ 32 ] with permission from Taylor and Francis Group, LLC The strength of nearly all ceramic materials decreases as the temperature increases as shown in Figure 10 It can be seen that the strength of ceramics changes only slightly for several hundred degrees up to a temperature where the strength decreases significantly. This appears to occur for most ceramics materials at intermediate temperatures. At higher temperatures, the rate of strength decrease is more rapid, generally attributed to non elastic effects [ 3 3 ] For a wide range of temperature, CaO exhibit only small temperature dependent changes in the yield strength up to 1100
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26 In the FGM beam models to be analyzed in Chapter 5 basically two materials will be used ; these are steel and alumina (A l 2 O 3 ) The thermal properties for the materials are shown in Table 2 and Table 3 These material property data w ere collected from engineering manuals ( [ 34 ] [ 35 ] ), material handbooks ( [ 36 37 ] ), and a n online database of material properties [ 38 ] Table 2 Thermal properties of steel [ 31 35 39 ] (W/m K ) (GPa) (1/ K ) (MPa) 0 61.8 194 11.4 4 20 27 60.7 204 11 6 39 7 100 57.8 195 12.1 38 1 200 53.5 20 4 12.7 362 300 49.0 193 13.3 380 400 44.5 188 13.9 359 500 40.2 18 3 14.4 31 3 600 35.7 167 14.8 284 700 31.2 14 1 15.0 167 800 27.3 10 6 14.8 72 900 26.0 74 12.6 44 Table 3 Thermal properties of alumina [ 31 38 ] (W/m K ) (GPa) (1/ K ) ( M Pa) 0 50.45 41 5 4 7 5 459 27 42.00 408 5 55 45 5 100 29.51 393 6 8 6 44 2 200 21.5 6 3 80 7 42 424 300 16.92 373 7 7 9 40 7 400 13.54 37 1 8 15 390 500 10.6 2 370 8 4 3 375 600 8.77 368 8 7 2 363 700 7. 80 36 4 9 02 35 5 800 7.08 353 9 29 350 900 6.4 5 33 6 9 53 349
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27 Figure 11 and Figure 12 p lot the temperature dependence of these two materials We ca n observe that for both materials the therm oelastic properties var y sign ificantly with the temperature fo r a wide range of temperature ( 0 o C 9 00 o C) which confirms the behavior discussed previously for several engineering materials in Figure 7 10 Also, from these figures it can be seen that the properties are not linear functions of the temperature and in general have large variations with temperature Therefore, a cubic splines interpolation is used to fit a model for the temperature dependent material property data of these two materials During the solution of the FGM beam model problems to be analyzed in Chapter 5 the se fitted models are incorporate d into the numerical procedure for solving these problems. The solution details of these models are treated in the two dimensional numerical s olution for a 3 l ayer FG Beam section in Chapter 4.
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28 (a) (b) (c) (d) Figure 11 Temperature dependence of the thermoelastic properties of steel
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29 (a) (b) (c) (d) Figure 12 Temperature dependence of the thermoelastic properties of alumina.
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30 FGMs Applications As technology progresses at an ever increasing rate, the need for advanced capability materials becomes a priority in the engineering of more complex and higher performance systems. This need can be seen in many fields in which engineers are exploring the applications of these new engineered materials. Aerospace engineers trying to incorporate new and improved capabilities into air and space systems are pushing the envelope for what current materials can physically handle. FGMs are a relatively new technolo gy and are being studied for the use in components exposed to harsh temperature gradients. There is an extensive variety of applications in engineering practice which require s materials performance to vary with locations within the component One of these applications is shown in Figure 13 where FGM is used to improve the thermo mechanical performance of a turbine blade design Another application, is shown in Figure 14 where graded region ( FGM ) is inserted between a metal and a ceramic tip for relaxation of stress concentration in lathe bits
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31 Figure 13 FGM application for a turbine blade design Reprinted from Ref. [ 40 ] with permission from Elsevier. Figure 14 FGM application for relaxation of stress concentration in lathe bits Reprinted from Ref. [ 41 ] with permission from Elsevier. There are many more current and future applications for FGMs Most of them include space shuttle s and aeronautical applications. One of these is the aircraft exhaust wash structures which separate exhaust gas from aircraft structure for vehicles which have internally exhausted engines, that is stealth aircraft and UAVs (Unmanned Aerial Vehicles) with engines that do not exhaust directly to the atmosphere. FGMs are also being used in the thermoelectric devices for energy conversion and the semiconductor industry. FGMs are also being used as thermal barrier coating in gas turbine engines [ 42 ] As research into this material progresses and the cost for manufacturing decreases, it is inevitable that many other industries which deal with severe thermal gradients will begin investigating the usefulness of FGMs.
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32 Chapter 4 Formulation of Governing Equations Introduction This chapter presents the fundamental aspects of the b eam t heory for analyzing FGM structures that serve s as a bas is for developing this research. D etailed formulation s of the g overning e quations for analyzing functionally graded material models are presented Formulations of the equations of motion are developed for a first order shear deformable beam It includes t wo finite element formulations The first element formulation is a two node formulation based on the work of Chakraborty et al [ 1 ] The second formulation is a new three node beam element formulation t o study the thermo elastic behavior of functionally graded beam structures Beam Theory for FGM Structures The axial and transverse displacements using first order shear deformation theory for a beam are given by ( 3 )
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33 ( 4 ) where and are the mid plane axial and transverse displacements, is the rotation about y axis and z is measured from the reference plane as shown in Figure 15 Using Eq s ( 3 ) and ( 4 ) and adding the strain due to temperature, the linear strains displacement relations are: ( 5 ) where ( ) x represents differentiation with respect to x ( z ) is the coefficient of thermal expansion, T is the temperature change from a stress free state. In general, the temperature can vary along the length and through the thickness. The constitutive relations are given by ( 6 ) where E ( z ) and G ( z ) s hear modulus respectively and are allowed to vary through the thickness. The strain energy S is given by ( 7 )
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34 Figure 15 Beam coordinate system. where L and A are the length and the area of cross section of the beam, respectively. Eq. ( 7 ) can also be written as ( 8 ) Using Eq. ( 5 ) the strain energy S is expressed in terms of displacement field as ( 9 ) Integrating over the area and, from composite laminate theory, the following stiffness and thermal stiffness coefficients are X Y Z h BL Reference Plane A width
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35 ( 10 ) ( 11 ) Using the relations in Eqs. ( 10 ) and ( 11 ) Eq. ( 9 ) can be rewritten as ( 12 ) Taking the first variation with respect to the nodal degrees of freedom and ( 12 ) into the equations of motion in terms of , and as : ( 13 ) : ( 14 ) : ( 15 )
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36 The force boundary conditions obtained from the line integral are ( 16 ) ( 17 ) ( 18 ) where and denote axial force, shear force and bending moment acting at the boundary. Note that when the temperature is constant along the axis of the beam, temperature does not enter into the governing equations but enters as a fo rce term in the boundary conditions Finite Element F ormulations Now that the equations of motion are developed for a first order shear deformable beam, the following two sections will develop the element formulation. The first formulation is based on Chakraborty et al. [ 1 ] followed by the three node element formulation. Two node E lement Formulation Chakraborty et al [ 1 ] obtained the same governing equations expressed in Eqs. ( 13 ) ( 15 ) However, the y developed the element for a constant temperature case only.
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37 Herein, the derived element formulation contains provisions for axial and through the thickness temp erature gradients. For clarity, the variables in Chakraborty et al [ 1 ] have been retained where possible. The element formulation starts by developing the interpolation functions based on the displacement field. The exact form for the degrees of freedom used in the general solution of Eqs. ( 13 ) ( 15 ) are ( 19 ) ( 20 ) ( 21 ) Each node in an element has three degrees of freedom and there are two nodes per element giving a total of six unknowns per element. A review of Eqs. ( 19 ) ( 21 ) gives ten unknown coefficients, Static condensation [ 12 ] [ 13 ] is employed to reduce the number of unknowns to six. Substituting Eqs. ( 19 ) ( 21 ) into Eqs. ( 13 ) ( 15 ) yields the following system of equations ( 22 ) ( 23 ) ( 24 )
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38 Solving the system of equations ( 22 ) ( 24 ) the following relationships are established ( 25 ) where , ( 26 ) In Eq. ( 26 ) and relate to coupling between the stiffness coefficients while and give the coupling between stiffness coefficients and the axial gradient of the th ermal stiffness coefficients. If the axial gradient is zero, the terms and are both equal to zero. With the aid of the relationships establishe d from static condensation, Eqs ( 19 ) ( 21 ) are rewritten in the form
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39 ( 27 ) ( 28 ) ( 29 ) Note that the end result of static condensation is a coupling between the mid plane displacements and mid plane rotations through the stiffness coefficients and the gradient in the axial direction of the thermal stiffness c oefficients. Eqs. ( 27 ) ( 29 ) can be rewritten in matrix form ( 30 ) where and Solving for the unknown constants, in terms of nodal variables requires evaluating Eq. ( 30 ) at the nodes and as (see Figure 16 )
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40 Figure 16 Nodes and degree s of freedom for the 2 node element ( 3 1 ) Rearranging and using a more compa ct notation Eq ( 3 1 ) is written as ( 32 ) w here Solving for the unknown constants , yields ( 33 )
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41 The term is the nodal displacement vector for the element and is the thermal gradient contribution vector at nodes. Now the displacements at any point in the element can be expressed in terms of nodal displacements by substituting Eq. ( 33 ) into Eq. ( 30 ) ( 34 ) Recognizing that is equal to the shape function for the element, The full expression for is given in Appendix A of Ref. [ 1 ] T emperature gradient in the axial direction results in and terms The terms and are zero when the beam does not contain a thermal gradient in the axial direction The ele ment stiffness matrix is determined by ( 35 ) where i s the strain displacement matrix. Performing the matrix multiplication, integrating over the volume, and using the definitions in Eq. ( 10 ) gives the terms of the element stiffness matrix in closed form. The element stiffness matrix is given in Appendix A of Ref. [ 1 ]
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42 The final step is formulating the t hermal load vectors. The first thermal load vector is from the force boundary conditions in Eqs. ( 16 ) ( 18 ) ( 36 ) The second thermal load vector comes from Eqs. ( 13 ) ( 15 ) Specifically the terms a nd c an be treated as distributed lo ads and applied at the nodes as ( 37 ) Eq. ( 37 ) can be numerically integrated for a general temperature distribution. If the temperature distribution is linear along the length of the beam, and a re constant s and the integration of Eq. ( 37 ) results in ( 38 ) w here
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43 For this special case, the thermal load vector i s constant along the beam length and we have the closed form of given by Eq. ( 38 ) However, f or a more general form of t he temperature distribution the terms and are no longer constants and it is necessary to numerically integrate Eq. ( 37 ) for each element of the beam model When is assembled globally for the model, the second and fifth terms cancel for all but the end elements. Finally, the full expression for the element with a general thermal load can be expressed as ( 39 ) Three node FOSD E lement In developing the three node element, the same d isplacement field given in Eqs. ( 19 ) ( 21 ) is used. By choosing an interpolation function for one order in x higher than m eets one of the requirements to prevent shear locking of the element [ 3 ] Static condensati on is again used to reduce the number of unknown coefficients from ten to nine. The relationship is found to be ( 40 )
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44 coupling the transverse displacement and the rotation. However, unlike in the two node element formulation the stiffness coefficients and the gradient of the thermal stiffness coefficients do not make an appearance in the interpolation functions and therefore in the shape functions. Proceeding as in the two node element case to so lve for the unknown coefficients in terms of nodal variables we evaluat e Eq. ( 29 ) at the nodes a nd (see Figure 17 ) the following matrix is written ( 41 ) or i n similar notation as Eq. ( 32 ) ( 42 ) Figure 17 Nodes and degrees of freedom for the 3 node element
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45 The shape function for the three node element becomes ( 43 ) where t he non zero elements of the shape function for the three node element are given by ( 44 ) The thermal load vectors for the three node element are arrived at through a similar process as in the two node element formulation case finding that ( 45 )
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46 a nd ( 46 ) Again, as in the 2 node element case, this value of in Eq. ( 46 ) is valid only for the particular case when the temperature distribution thermal load is linear. For a general form of t he temperature distribution the terms and a re no longer constants and it is necessary to numerically integrate Eq. ( 37 ) for each element of the beam model. Next we find t he element stiffness matrix for the three node element. Proceeding in similar way as in the two node element case, t he nonzero elements in the upper diagonal of the stiffness matrix are found to be : ; ;
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47 ; ; ; ; ( 47 ) ; ; ; ;
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48 Temperature P rofile M odeling This section presents the mathematical formulation and solution of the heat conduction steady state problem for composite FGMs beam models under thermal loading The solution serve s as a foundation to conduct the analytical and numerical simulations in this research. Two formulatio ns are presented. The first is a formulation to find the one dimensional temperature distribution for a 3 layer beam with a middle FG M layer This is fo llowed by a more realistic numerical formulation for finding the two dimensional temperature distribution for the same 3 layer beam One d imensional Heat Conduction Steady State Exact Solution for a 3 layer FG Beam In this part we consider the solution of the heat conduction steady state problem in a composite beam consisting of 3 layers, which are assumed to be in perfect thermal contact. Figure 18 shows the geometry coordinates and the boundary conditions for this problem.
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49 Figure 18 Three layer beam with perfect thermal contact at the interface. The mathematical formulation of this problem is given as ( 48 ) ( 49 ) ( 50 ) subject to the boundary and interface conditions ( 51 ) z 0 a h 2 a h 1 T 3 (z) T 2 (z) T 1 (z) T 3 = T t T 1 = T b Graded Layer x Homogeneous Layer Homogeneous Layer
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50 ( 52 ) ( 53 ) ( 54 ) ( 55 ) ( 56 ) where , and are the thermal conductivities coefficients for steel, graded layer, and alumina, respectively (see Figure 19 ) The solution to the equations ( 48 ) ( 50 ) subject to the boundary and interface conditions given by Eqs. ( 51 ) ( 56 ) can be found numerically. Several special cases can result in exact solutions such as when and are constant throughout layers 1 and 3 while is assumed to vary only in the direction of the beam thickness according to ( 57 ) The solution of the o rdinary differential equations ( 48 ) ( 50 ) for each layer is given in the form ( 58 ) ( 59 ) ( 60 )
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51 The solution involves two unknown constants for each layer; then, for a 3 layer problem, 6 unknown constants are to be determined. Substituting the solution given by Eqs. ( 58 ) ( 60 ) into the boundary and interface conditions ( 51 ) ( 56 ) one obtains 6 equations for the determination of the 6 unknown constants. T he final solution for each layer is then given by ( 61 ) ( 62 ) ( 63 ) Figure 19 shows the depth wise exact temperature distribution for sample material s and geometrical parameters given in Table 4 Table 4 M aterial and geometrical parameters of a tri layered b eam Parameter Value h 1 0.0 25 m h 2 0.02 5 m a 0.025 m T b 2 0 o C T t 4 00 o C k b 51.9 W/m o C ( Steel ) k t 13.75 W/m o C ( Al 2 O 3 )
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52 Figure 19 Depth wise exact temperature distribution obtained from the solution of the heat conduction differential equation.
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53 Two dimensional Heat Conduction Steady State Numerical Solution for a 3 L ayer FG Beam with Temperature Dependency of the Material Properties Now we present the mathematical formulation of the two dimensional heat conduction steady state problem of a composite beam consisting of three parallel layers which are assumed to be i n perfect thermal contact. Figure 20 shows the geometry coordinates and the boundary conditions for this problem. Different from the one dimensional profile modeling case, in this problem the material parameter thermal conductivity depends on the temperature itself, which is the dependent variable of this problem. Figure 20 Three layer beam geometry and boundary conditions
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54 The two dimensional mathematical formulation of this problem is given as ( 64 ) ( 65 ) ( 66 ) subject to the boundary and interface conditions ( 67 ) ( 68 ) ( 69 ) ( 70 ) ( 71 ) ( 72 ) ( 73 ) ( 74 )
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55 where , and are the thermal conductivities coefficients for steel, graded layer, and alumina, respectively Since layers 1 and 3 are homogenous materials, their thermal conductivities and are considered independent of the position z throughout layers 1 and 3 ; however they still depend on the temperature The thermal conductivity of the graded layer is assumed to vary in the direction of the beam thickness and with the temperature according to ( 75 ) Th e partial differential equations ( 64 ) ( 66 ) are classified as elliptic type [ 43 ] This kind of equation is also well known as the homogeneous Laplace equation It is important to realize that this problem be come s nonlinear due to the nonlinearity introduced to the governing differential equation by the variation of the thermal conductivity with the temperature which is the dependent variable itself of this problem This nonlinear heat conduction steady state problem was solved iteratively using a finite element partial differential equation solver using the computational tool MATLAB During the solution, the temperature dependency of the thermo mechani cal material properties is considered. That is, during the solution process of finding the temperature distribution in the different layers these material properties are update d iteratively according to the actual temperature at the particular geometrical position. The material property data was fitted using cubic spline interpolation, as discussed in the temperature dependence of material properties section in Chapter 3 and incorporate d into the
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56 numerical procedure so that the solver can interpolate to determine the thermal conductivity of a material at any temperature. The resulting temperature distribution will be used as a thermal load into a finite element code for analyzing stress es in FGM beam models in Chapter 5.
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57 Chapter 5 Analys e s and Results Introduction This chapter discusses the results of the analyses performed in this research. The analyses are performed using the computational tool MATLAB C omparisons with FGMs model s available in related literature are made Also, a nalytical and n umerical s imulations of t hermal loading studies conducted on the FGMs beam structures are presented The beam models are studied to show the performance of the element formulations presented in Chapter 4 The mathematical formulation and solution details of th ese problems are included in Chapter 4 as well. A dditionally this chapter introduces a study to determine the influence of manipulating the FGM layer thicknesses on the f actor of s afety in structures constructed of functionally graded materials under therm al loads Comparisons of the Element Formulation Simulations with Related Literature This section will present simulations of FGMs model results available in the related literature for comparison purposes. These comparisons will reveal the performance of the element formulations presented in Chapter 4 Two groups of
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58 comparisons will be presente d. The f irst group involve s an example of thermal stress distribution in a tri layered FGM model analyzed by Suresh and Mortensen [ 2 ] The s econd group of comparison involves the analysis of FGM beams in stress smoothening when more than one type of material is present in the structures as presented by Chakraborty et al [ 1 ] For both groups, problems will be revisited and their results will be compared with the results of this research [ 1 2 ] Comparison with Suresh and Mortensen M odel The formulated element in this work is used to compare with an example of thermal stress distribution in a tri layered FGM model analyzed by Suresh and Mortensen [ 2 ] The model considered is a system of Ni graded layer (GL ) Al 2 O 3 tri layered composed beam as shown in Figure 21 In this literature the thermoelastic properties within the graded layer vary linearly with z according to ( 76 ) ( 77 ) where preceding a property refers to the change in that property for a change in temperature and the subscript on a property refers to the value of that property at the initial reference temperature. At this point, it important to mention that e ven though
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59 this literature uses a different approach for calculating the material properties and does not give details ab out the actual properties values used in its model, our intention here is to make a qualitative comparison rather than a quantitative one. With this in mind, we proceed to compare the results for the a xial thermal stress distribution throughout the thickne ss found in the referenced literature and the present work. Figure 21 Geometry and nomenclature for a tri layered composed beam model from literature reference [ 1 ] Figure 22 shows the spatial variation of the thermal axial stress (in plane stress) throughout the thickness of the Ni GL Al 2 O 3 tri layered beam subject to a temperature drop of 100 o C (from an initial stress free reference temperature) for the particular geometrical condition that and that The constituent m aterials of this mod el and their properties are given in Table 5
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60 Table 5 Thermo elastic p roperties of nickel and alumina at 300 K Property Parameter v alue Nickel Al 2 O 3 odulus of elasticity 2 07 GPa 390 GPa Shear modulus 76 GPa 137 GPa Coefficient of thermal expansion 1 3 1 10 6 o C 1 6 9 10 6 o C 1 When comparing the axial stress distribution found in the present work shown in Figure 22 (a) with that in the reference literature ( Figure 22 (b)), the following tendencies are reveal ed : 1. when there is no graded layer between the ceramic and metal layer, large values of stresses are developed at the interface; 2. the near interface region of the metallic layer is in tension, while the corresponding region for the ceramic layer is in compression (there is considerable abrupt change in magnitude and sign of the stress at the interface); 3. when a graded interlayer is introduced, the magnitude of the stress at the interface can be significantly reduced and the abrupt change in the stress sign is eliminated; 4. the stresses vary linearly with z within the metallic and ceramic layer, and approximately parabolically within the functionally graded layer. From these trends, we can conclude that the qualitative results obtained in this work are very similar to the referenced literature which demonstrate s a suitable performance of the element formulations presented in this work
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61 a) P resent work results b) Reference literature Figure 22 Axial t hermal s tress distribution in a Ni Graded Layer Al 2 O 3 trilayer beam subject to a T = 100 o C Figure 22 (b) reprinted from Suresh and Mortensen [ 2 ] with permission from Maney Publishing
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62 Comparison with Chakraborty et al. M odels Next, we compare t he axial and shear stress through the thickness results obtained with formulated element in this work with the results obtained in Ref. [ 1 ] for a bi material beam model where the transition is made smooth by inserting a thin FGM layer. The m ate rials considered and their properties are given in Table 6 Using these materials a functionally graded cantilever composite beam of 0 5 m length and unit width subjected to three different loads are considered as illustrated in Figure 23 The topmost material is steel and bottom layer is alumina. A n FGM inter layer is placed i n between these layers. Material properties vary according to the exponential law given by ( 78 ) where describes a typical material property ( , etc.) at any point throughout the thickness T he variables and are the material property at the top and bottom, respectively Table 7 specify the l oading cases applied to the analyzed models Each of these loads is applied to three different geometrical configuration s of the two material constituent of this model. The first is a bi material beam contains two layers; t he second is a partial functional graded composite beam (PFGM) consisting of 3 layers where the m iddle region is a FGM that transitions the material properties from the bottom layer to
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63 the top layer; t he last a 1 layer beam composed of a functional graded material (FGM) through the entire thickness a) unit transverse load b) unit axial load c) uniform thermal load Figure 23 Geometry and loading cases for models from literature paper [ 1 ] Table 6 Thermo elastic p roperties of steel and alumina at 300 K Property Parameter v alue Steel Al 2 O 3 Modulus of elasticity 210 GPa 390 GPa Shear modulus 80 GPa 137 GPa Coefficient of thermal expansion 14.0 10 6 o C 1 6 9 10 6 o C 1 Table 7 Loading cases for m odel s from literature paper [ 1 ] Case Load type 1 Unit transverse load applied at the tip (1 N) 2 Unit axial load applied at the tip (1 N) 3 T hermal load T = 5 o C
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64 The results of this comparison are summarized in Figure 24 28 From the F igu re s 24 28 we can observe the following similarities and discrepancies w hen comparing this work and the referenced literature results Both results agree as follows : 1. in the absence of FGM layer between the ceramic and metal layer the stress distributions are discontinuous at the interface; 2. the i ntroduction of a small FGM layer smoothens the stress es to the tune of about 300 N / m 2 and 10 N / m 2 stress jump of the axial and shear stress respectively (the abrupt value change in the stress is eliminated) ; 3. for load cases 1 and 2, the axial stresses vary linearly with z within the metallic and ceramic layer, and approximately parabolically within the functionally graded layer; 4. for lo ad cases 1 and 2, the shear stresses are constant throughout the metallic and ceramic layer, and approximately parabolically within the functionally graded layer. The results disagree in the axial stress for the load case 3 (thermal load). In the present w ork the axial stress vary linearly with z within the metallic and ceramic layer, while for the referenced paper the axial stress is constant throughout these layers. From these observations we can conclude that except for thermal load case, the qualitative and quantitative results obtained in this work are very similar to the referenced paper which demonstrate s a proper performance of the element formulations presented in this work.
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65 a) P resent work results b) Reference paper Figure 24 A xial stress through the thickness for case 1 Figure 24 (b) r eprinted from Ref. [ 1 ] with permission from Elsevier a) P resent work results b) Reference paper Figure 25 T ransverse shear stress through the thickness for case 1 Figure 25 (b) r eprinted from Ref. [ 1 ] with permission from Elsevier
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66 a) P resent work results b) Reference paper Figure 26 A xial stress through the thickness for case 2 Figure 26 (b) r eprinted from Ref. [ 1 ] with permission from Elsevier a) P resent work results b) Reference paper Figure 27 A xial stress through the thickness for case 3 Figure 27 (b) r eprinted from Ref. [ 1 ] with permission from Elsevier
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67 a) P resent work results b) Reference paper Figure 28 T ransverse shear stress through the thickness for case 3 Figure 28 (b) r eprinted from Ref. [ 1 ] with permission from Elsevier
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68 Simulations w ith G eneric T emperature D istributions and T emperature I ndependence of the M aterial P roperties This section analyzes the beam configurations shown in Figure 29 The model s are composed of a cantilever beam with the support at the origin. The beam is 100 mm long and 10 mm thick. The b eam width is not important because in the first order shear deforma tion theory for a beam this is classified as cylindrical bending. a) FGM beam b) Bi material beam c) Bi material with average interlayer beam d) PFGM beam Figure 29 Beam configurations. The beam contains two materials arranged in four different configurations The first Figure 29 (a) is a 1 layer beam composed of a functional graded material (FGM) through the entire thickness. T he second Figure 29 (b) is a bi material beam contains two X Y Z h BL Reference Plane width X Y Z h BL Reference Plane width X Y Z h BL Reference Plane width X Y Z h BL Reference Plane A width
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69 layers with out FGM region similar to a traditional layered composite The third one Figure 29 (c) is a composite beam consisting of 3 parallel layers where the middle region is a homogeneous layer wh ose properties values are the average of the material properties of the bottom layer ( homogeneous material) and the top layer (homogeneous material). The last Figure 29 (d) is a partial functional graded composite beam (PFGM) consisting of 3 parallel layers where the middle region is a FGM that transitions the material properties from the bottom layer (homogeneous material) to the top layer (homogeneous material). The beam models are subject to different thermal loads using generic temperature distribution s (some of them from the related literature) ; its mechanicals and thermal properties are independent of temperature The FGM beam has a modulus ratio, E t / E b of 5 with E b equal to 1GPa. The ed at 0.3125. The coefficient of thermal expansion has a ratio of t / b of 1/5 with t equal to 10 4 Within the FGM region, the thermo mechanical material propert ies, the modulus thermal conductivity and coefficient of thermal expansion vary through the thickness following the corresponding formula presented in Table 1 of Chapter 3 Figure 30 shows details about the beam geometry and boundary conditions of the different models to be an a lyzed The PFGM beam contains three sections where the middle region is a FGM that transitions the material properties from section 1 to section 3. Sections 1 and 3 are a quarter of the beam thickness with a constant modulus and coefficient of thermal expansion equal to the bottom and top of the FGM example respectively. The bi material beam contains two sections of equal thickness with no FGM region similar to a traditional layered composite.
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70 a) FGM beam b) Bi material beam c) Bi material with average interlayer beam d) PFGM beam Figure 30 Beam geometry and boundary conditions. The temperature distributions applied to the models are summarized in Table 8 The temperature distributions are with reference to stress free configuration. Case 1 represents a constant temperature. Case 2 contains a thermal gradient in the through the thickness only. Case 3 contains an axial thermal gradient. Case 4 combines cases 2 and 3 to give a distribution with an axial and through the thickness temperature gradient. z 0 a h 2 a h 1 T 3 (x,z) T 2 (x,z) T 1 (x,z) T 3 = T t T 1 = T b Graded Layer x Homogeneous Layer Homogeneous Layer z 0 a h 2 a h 1 T 3 (x,z) T 2 (x,z) T 1 (x,z) T 3 = T t T 1 = T b Average Layer x Homogeneous Layer Homogeneous Layer z 0 h 2 h 1 T 2 (x,z) T 1 (x,z) T 2 = T t T 1 = T b x Homogeneous Layer Homogeneous Layer 0 a a T 1 (x,z) T 1 = T t T 1 = T b Graded Layer x z
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71 Table 8 T emperature distributions. Case T ( x,z ) C 1 1 2 10 3 1 4 10 In the first example, a constant temperature distribution is applied to the beam and the normalized axial stress through the normalized thickness is shown in Figure 31 Both element formulations yield nearly identical results for all four beam material combinations. The combination of boundary conditions and thermal loading produces no gradients in the axia l direction so Figure 31 applies along the length of the beam. The transverse shearing stresses are equal to zero for this load case. The bi material beam contains the highest peak stress followed by the PFGM and FGM as expected showing the advantage of smooth as opposed to discontinuous transitions in material properties. Only at the bottom surface is the stress in FGM greater than either the PFGM or b i material beam.
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72 Figure 31 Normalized axial stress through the thickness for case 1, T =100. The stress is normalized by ( E b b T) and transverse coordinate (z) is normalized by the beam thickness. The second example contains an exponential through the thickness variation in temperature. The temperature change at the top of the beam is 1000 C and the bottom is 100 C from a stress free state temperature. The normalized axial stress in the transverse direction is shown i n Figure 32 The combination of boundary conditions and thermal loading produce a beam whose stress is invariant to the axial direction. The transverse shear stress is equal to zero as well. In terms of comparing the two formulations, both give identical results.
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73 Figure 32 Normalized axial stress through the thickness for case 2, T ( z ) = The stress is normalized by ( E b b T b ) and transverse coordinate (z) is normalized by the beam thickness. The next case contains a linear axial temperature distribution from 0 C at the cantilevered end to 100 C at the free end. Figure 33 displays the normalized axial stress through the thickness of the beam. The normalized stress does not vary along the length of the beam. Case 1 and 3 are very similar and only show minor differences in the normalized axial stress. However, transverse shear stress is present as shown in Figure 34 when using the equilibrium equation but zero when calculated from the constitutive relations. The 2 node beam formulation gives poor results while the three node beam formulation gives acceptable results. This is due to the differences in the second deriv ative of the shape function which is used to calculate the shearing stress from the equilibrium equation.
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74 Figure 33 Normalized axial stress through the thickness for case 3, T ( x ) = The axial stress, is normalized by ( ) and transverse coordinate (z) is normalized by the beam thickness. Figure 34 Normalized transverse shear stress through the thickness for case 3, T ( x ) = The shear stress, is normalized by ( ) and transverse coordinate (z) is normalized by the beam thickness.
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75 For a traditional two node beam element the first derivative with respect to x of the shape function is constant and the second derivative is a zero matrix. Because of the coupling terms from static condensation, the second derivative of the shape function for the two node beam element formulated in the preceding chapter is not the zero matrix. However, the resulting shear stress when using this matrix is undesirable and gives a maximum shearing stress at the top of the beam which should be zero. In contrast, the three node elem ent formulation gives a reasonable result. It should be emphasized that the magnitude of the transverse shear stress is small and is given on the y axis on the right hand side of the graph for the three node element. The absolute value obtained from the di fference between the normalized axial stress of case 1 and case 3 is of the same order as the absolute value of the transverse shear stress. The final case is a combination of cases 2 and 3 which gives a temperature distribution with a gradient in the axia l and transverse thermal direction. The normalized axial stress in the though the thickness direction is shown in Figure 35 and the corresponding transverse shear stress using the equilibrium equation in Figure 36 The transverse shear stress from the constitutive relations is zero. The normalization procedure produces a graph independent of the axial location. The axial stress in this case does not vary significantly from case 2 with the same transverse variat ion in temperature. Again both elements produce nearly identical results for the axial stress but vary greatly when considering the transverse shearing stress.
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76 Figure 35 Normalized axial stress through the thickness for case 4, T ( x,z ) = The axial stress, is normalized by ( ) and transverse coordinate (z) is normalized by the beam thickness. Figure 36 Normalized transverse shear stress through the thickness for case 4, The shear stress, is normalized by and transverse coordinate (z) is normalized by the beam thickness.
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77 Simulations with A ctual T emperature D istributions with and without T emperature D ependence of the M aterial P roperties This section analyze s the examples considered in the preceding section using the actual temperature distribution found by solving the heat conduction steady state problem for the different composite beam models under thermal loading. The material properties law, assumptions, geometry and layers configuration are as per the preceding section. The temperature distribution was found as dimensional H eat C onduction S teady S tate Numerical S olution for a 3 L ayer FG Beam in Chapter 4 Also, the examples are analyzed considering the temperature dependency of the thermo elastic material properties. The temperature dependent material property data was collected from engineering manuals, material handbooks, and database of material properties web sites [ 31 35 39 ] The material property data was fitted using cubic spline i nterpolation and incorporated into the numerical procedure. T he nonlinear heat co nduction steady state problem was solv ed iteratively using a finite element solver. The solutions details of this problem are given in the two dimensional heat conduction steady state problem section in Chapter 4. The following figures show boundary conditions, thermal conductivity and temperature profiles for the analyzed models. The first analyzed model is a two layer beam composited of steel and alumina as shown in Figure 37 This model will serve as a baseline reference to compare how the thermal conductivity temperature dependence affects the temperature distribution and thermal stresses. It also will reveal how the temperat ure distribution and thermal stresses behaves when varying interlayers are introduced.
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78 Figure 37 Beam geometry and boundary conditions (Bimaterial) Figure 38 shows thermal conductivity distribution with and without temperature dependence. Figure 38 ( a ) shows that the thermal conductivity is constant throughout the entire layer for each material ( 51.26 W/m K for steel and 18.41 W/m K for alumina both at at 235 o C ) when temperature influence is not consider ed However, Figure 38 ( b ) reveals the actual thermal conductivity distribution when temperature dependence i s take n into account. We can observe that when temperature d ependen ce i s considered the previously assumed constant thermal conductivities values for steel and alumina vary from about 45 to 62 W/m K for steel and from about 12 to 40 W/m K for alumina. The significance or effect of this observation can be seen in the temperat ure profile distribution shown in Figure 39 It can be seen that at a particular position z other than a boundary, the temperatures are higher in Figure 39 (a). In other words, the heat insulation effect of alumina is higher when the temperature dependence is considered. This can be explain ed by the fact that as the temperature increase s, the thermal
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79 Consideration of Temperature Chapter 3 ( see Figure 11 and Figure 12 ) (a) (b) Figure 38 Thermal conductivity distribution with and without temperature dependence (Bimaterial case) (a) (b) Figure 39 Temperature profile with and without temperature dependence (Bimaterial case).
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80 The next analyzed model is a three layer beam composited of a steel bottom layer, an alumina top layer, and a homogeneous material interlayer, as shown in Fig ure 40 The material properties of this interlayer are taken as the average values of the steel and alumina. Figure 41 shows thermal conductivity distribution with and without temperature dependence for this model. Again, we can see from Figure 41 (a) that the thermal conductivity is constant throughout the entire layer for each material (5 1.26 W/m K for steel 18 .41 W/m K for alumina and 34.83 W/m K for the average interlayer ; proper ties are taken at the average temperature 235 o C ) when temperature influence is not considered As in the preceding bi material model the actual thermal conductivity distribution for this model is very different when temperature dependence is taken into a ccount ( Figure 41 (b) ) Here w e can observe that when temperature dependence is considered the actual thermal conductivities values for steel alumina and the average interlayer vary from 55 to 62 W/m K 12 to 40 W/m K and 30 to 50 W/m K respectively Again, as in the bi material model this difference on the thermal conductivity distribution a ffect s the temperature profile distribution as shown in Figure 42 Similar behavior in comparison with the bi material model can be seen here. That is, for a particular position z other than a boundary, the temperatures are higher in Figure 42 (a). Also in this model, it is found that the heat insulation effect of alumina in Figure 42 (b) is hi gher than in Figure 42 (a). Now, comparing this model with the bi material model, we see that in this model the heat insulation effect of alumina is higher
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81 Fig ure 40 Beam geometry and boundary conditions (Average interlayer)
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82 (a) (b) Figure 41 Thermal conductivity distribution with and without temperature dependence ( Average interlayer case). (a) (b) Figure 42 Temperature profile with and without temperature dependence ( Average interlayer case).
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83 Now, we study the effect of substituting the homogeneous material interlayer by a functionally graded material (FGM) interlayer, as shown in Figure 43 The material properties of this interlayer are calculated according to the formulas given in Table 1 in Chapter 3 Figure 43 Beam geometry and boundary conditions ( FGM interlayer ) Figure 44 depicts the thermal conductivity distribution with and without temperature dependence for this model. For this model, we can see from Figure 44 (a) that the thermal conductivity is constant throughout the entire layer for the homogenous layers (51.26 W/m K for steel and 18.41 W/m K for alumina both at 235 o C ) but it change s continu ously from 51.26 to 18.41 W/m K for the FGM interlayer when temperature influence is not considered. In Figure 44 (b) we see that t he actual thermal conductivity distribution is very different f rom the results in Figure 44 (a) whe re
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84 temperature dependence was not considered For this model, the actual thermal conductivities values for steel, alumina, and the FGM interlayer vary from about 60 to 62 W/m K 12 to 40 W/m K and 20 to 6 0 W/m K respectively. The influence of this different behavior is manifested in the temperature profile distribution shown in Figure 45 As in the two preceding models, similar results are found for this model; t hat is, the temperatures are higher in Figure 45 (a) than in Figure 45 (b) for a particular position z other than a boundary Again, in this model the heat insulation effect of alumina in Figure 45 (b) is higher than in Figure 45 (a). When comparing this model with the bi material and average interlayer model s we see that in this model the heat insulation effect of alumina is higher. This fact can be use in engineering applications where insulation effects need to be improved.
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85 (a) (b) Figure 44 Thermal conductivity distribution with and without temperature dependence ( FGM interlayer case). (a) (b) Figure 45 Temperature profile with and without temperature depe ndence ( FGM interlayer case).
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86 T o compare the thermo elastic behavior of the three preceding models, we now analyze the thermal stresses and factor of safety in these models subjected to the corresponding temperature distribution found for each model. The results of this comparison are summarized in t he following figures The normalized axial stress through the thickness is shown in Figure 46 From Figure 46 (a) we observe that when temperature dependence is considered, the absolute value of axial stress is diminished at particular position z w ithin the steel layer For the FGM and ceramic layer s th e axial stress behavior is no n uniform As we can see, within these two last layers, there are regions where the absolute value of the axial stress is diminished when temperature dependence is considered and regions where the behavior is opposite. When comparing the influence of the temperature for the three analyzed model s it can be seen from Figure 46 (b) that the absolute value of axial stress is diminished in the average and FGM interlayer model withi n the steel layer. For the FGM and ceramic layers, the axial stress behavior is no n uniform. From these results, apparently nothing definitive can be conclude d yet regarding the influence of including the temperature dependenc y in the design of the beam. H owever, later on in this section we will see that the inclusion of the factor of safety and the specific factor of safety as a design criteria w ill allow us to chose the best design
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87 ( a ) (b) Figure 46 Normalized axial stress through the thickness for actual temperature distribution The stress is normalized by ( E b b T) and transverse coordinate (z) is normalized by the beam thickness.
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88 Figure 47 displays the transverse shear stress through the thickness of the beam. From Figure 47 (a), it is found that for the FGM interlayer model the absolute value of the shear stress diminishe s throughout the entire beam when temperature dependenc y is considered. When comparing the influence of the temperature for the three analyzed model s it can be seen from Figure 47 (b) that apparently the bi material model gives the lowest le vels of the absolute value of shear stress compared to the average and FGM interlayer model However, as commented for axial stresses results, we cannot make a final decision or conclusion regarding which model is better until we include the factor of safe ty and the specific factor of safety as design criteria.
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89 (a) (b) Figure 47 Normalized transverse shear stress through the thickness for actual temperature distribution The shear stress, is normalized by ( ) and transverse coordinate (z) is normalized by the beam thickness.
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90 As discuss ed earlier in this section, we calculate the factor of safety of the model s to have a decision criterion for finding the most conve nient beam model. Figure 48 displays the f actor s of safety and their corresponding position of calculation for the different analyzed models Table 9 summarizes the numerical values of these factors of safety. Figure 48 Factor of safety for the different models
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91 Table 9 Factor of safety for the different models Case Temperature independent Temperature dependent Bimaterial (No FGM interlayer) 1.8964 1.3521 Tri layer (Average interlayer) 1.6756 1.2869 Tri layer (FGM interlayer) 2.0445 1.1257 From the results in Table 9 we see that in general the factor of safety of the models decreases when temperature dependenc y is considered. Although the factor of safety is show n to decrease by adding an interlayer, these results are only for the special model case shown in Fig ure 40 and Figure 43 In a later section, we will show how a different three layer case gives higher factor of safety.
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92 Influence of the I nterlayer T hickness on the Factor of Safety in C omposite B eams This section describes a study to determine the influence of manipulating the FGM inter layer thickness of the beam on the f actor of s afety in structures constructed of functionally graded materials under therm al loads This study will allow among other benefits, an a nalysis/comparison of the advantages/benefits of using structures constructed of functionally graded materials with respect to those constructed with homogenous materials The beam models to be used in this study are shown in Figure 49 As noted, they are essentially the same composed cantilever beam s studied in previous sections The interest ed output s are the f actor of s afet y and the maximum temperature on the beams layers constructed of FGMs under therm al loads The finite element program developed in chapter 3 is used to automate th is study. a) Bi material beam b) PFGM beam Figure 49 B eam models for study ing the e ffect of the FGM i nterlayer t hickness in the factor of safety First, the bi material model is studied to find out the maximum thickness of the metallic layer able to meet the maximum temperature constraint in that layer The upper layer of the beam ( ceramic ) was allowed to be made thinner as the lower layer ( metallic ) was increased in thickness. Once this maximum possible thickness was found, it serve d z 0 a h 2 a h 1 T 3 (x,z) T 2 (x,z) T 1 (x,z) T 3 = T t T 1 = T b Graded Layer x Homogeneous Layer Homogeneous Layer z 0 h 2 h 1 T 2 (x,z) T 1 (x,z) T 2 = T t T 1 = T b x Homogeneous Layer Homogeneous Layer
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93 as baseline thickness of the metallic layer for studying the influence of the graded interlayer thickness on the f actor of s afety for the 3 layer composite beam. Deter mination of the Baseline Thickness of the Metallic Layer for Studying the Influence of the FGM Interlayer in the F actor of S afety For different metallic layer thicknesses the maximum temperature in the metallic layer was calculated for the bi material beam model (steel / Al 2 O 3 ). Without losing generality, t he maximum temperature allowable within the steel layer was set to 1 6 0 C Although the factor of safety is also calculated for the bi material models in this section, it was not used as a determin ing f actor in finding the baseline thickness of the metallic layer; it was included just to have a preliminary idea of its behavior when changing the layer thicknesses of the model. Table 10 Layer thickness variation for the bi material model. Steel thickness (m) Alumina Thickness (m) Max. temp. steel ( C) Factor of safety 0.0005 0.0095 27.90 1.5843 0.0010 0.009 0 36.16 1.7080 0.0015 0.0085 44.7 2 1.8692 0.0020 0.0080 53.7 3 1.8481 0.0025 0.0075 63. 20 1.8272 0.0030 0.0070 73.1 5 1.7871 0.0035 0.0065 83.8 1 1.7507 0.0040 0.0060 95.2 6 1.6345 0.0045 0.0055 1 07 .85 1.5002 0.0050 0.0050 120.92 1.3527 0.0055 0.0045 1 35 6 2 1.2936 0.0060 0.0040 1 51 8 4 1.2501 0.0061 0.0039 1 55 3 1 1.2235 0.0062 0.0038 1 58 86 1.2169 0.00621 0.00379 1 59 22 1.2166 0.00622 0.00378 1 59 58 1.2161 0.00623 0.00377 1 59 94 1.2157 0.00624 0.00376 160.31 1.2153
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94 The numerical results of this study are shown in Table 10 T he results reveal that as the steel thickness is increased and the ceramic layer thickness is decreased the maximum temperature in steel increases. Fr om the results we can establish that the baseline thickness of the metallic layer is 0.00623 m. Also, as a preliminary examination, we can see that the factor of safety of the beam tends to diminish as we reduce the ceramic material thickness from the beam. This fact gives us a criterion for choosing the placement of the FGM interlayer in next section.
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95 Effect of T hickness of the G raded I nterlayer in the F actor of S afety for the T ri layer M odel For different graded interlayer thicknesses the factor of safety and the specific factor of safe ty were calculated f or the 3 layer beam model (steel /FG/ Al 2 O 3 ). The numerical results of this study are shown in Table 11 T he maximum temperature allowable within the steel layer was set to 1 6 0 C For the FGM interlayer the maximum temperature is constraint to satisfy the following condition, based on the rule of mixtures, ( 79 ) where refers to the temperature in the FGM interlayer and and the volume fraction of the steel and ceramic layer, respectively. Regarding the placement of the FGM interlayer, we found in preliminar y computations of the tri layer model that the temperature constraints of the model do not allow the interlayer to be a replacement toward the ceramic layer. Based on this fact, we set the following conditions for this study: 1. we take the baseline thickness of the metallic layer found in the previous section (0.00623 m) as the maximum thickness of the steel to meet the maximum temperatu re requirement within it; 2. the baseline thickness of the ceramic layer was fixed at 0.0037 7 m; 3. the FGM layer thickness was increased in the direction of the steel layer, that is, toward the bottom boundary face diminishing the amount of steel from the mode l
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96 Table 11 Layer thickness variation for the 3 layer model. Steel thickness (m) FGM interlayer thickness (m) Alumina thickness (m) Max. temp. steel ( C) Max. temp. FGM ( C) Factor of safety Specific factor of safety 0.00623 0.0000 0.0037 7 1 59 94 1 59 94 1.2 2 0.19007 0.00 573 0.000 5 0.0037 7 1 46 18 1 63 35 1.1 9 0.18864 0.005 23 0.00 10 0.0037 7 1 32 9 6 1 66 39 1.1 9 0.19178 0.00 473 0.001 5 0.0037 7 1 20 2 3 1 68 78 1.2 1 0.19776 0.00 423 0.00 20 0.0037 7 1 08 1 5 1 72 02 1.22 0.20388 0.00 373 0.002 5 0.0037 7 96.4 3 1 74 27 1.25 0.21209 0.00 323 0.00 30 0.0037 7 85.0 6 1 76 94 1. 30 0.22319 0.00 273 0.00 3 5 0.0037 7 74.2 6 1 79 0 2 1.3 4 0.23373 0.00 223 0.00 4 0 0.0037 7 63.52 1 81 6 1 1.3 7 0.24311 0.00 173 0.00 4 5 0.0037 7 53.2 7 1 83 8 2 1.3 8 0.24977 0.00 123 0.00 5 0 0.0037 7 43.3 2 1 85 9 4 1.37 0.25339 0.00 073 0.00 5 5 0.0037 7 33.6 5 1 88 29 1.3 5 0.2531 0 0.00023 0.0060 0.0037 7 24.2 5 1 89 8 5 1.3 2 0.25263 0.00013 0.0061 0.0037 7 22.39 1 90 2 4 1.3 1 0.25174 0.00003 0.0062 0.0037 7 20.55 1 90 6 3 1.30 0.25116 Figure 50 shows the factor of safety as a function of FGM interlayer thickness From this figure we can see that the factor of safety of the beam tends to behave no nevenly as the interlayer thickness increases. Initially, for relatively low interlayer thicknesses (0 to 0.0010 m), the factor of safety decreas es then for thicknesses from 0.0010 to 0.0045 m it increases up to its maximum value of 1.3 8 For thicknesses between 0.0045 to 0.0062 m, the factor of safety starts decreasing its value again up to 1.30 As we can see, t his value of the factor of safety is not that low compared to those found for low interlayer thicknesses. It is important to realize that for small interlayer thicknesses the amount of metallic material in the beam is high, while for larger interlayer thicknesses the content of metal is low. From this fact, we can conclude that, in general, the factor of safety tends to improve as we i ncrease the FGM interlayer thickness in the beam.
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97 From Figure 50 and based in the factor of safety criterion, we could tend to decide that the best FGM interlayer thickness is 0.0045 m where its factor of safety has a maximum value of 1.38 However, as we discuss next we will see that this is not the best decision cr iterion. Figure 50 Effect of t hickness of FGM i nterlayer in the factor of safety for the tri layer model
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98 Figure 51 Effect of t hickness of FGM i nterlayer in the specific factor of safety for the tri layer model To make a better decision criterion for finding the best interlayer thickness, we use the specific factor of safety of the model, which is given by ( 80 ) where , and are the specific factor of safety factor of safety, a nd specific gravity of the beam respectively. The ratio is a convenient decision parameter in determining the interlayer thickness since it combines together the strength and mass of t he beam Figure 51 shows the specific factor of safety as a function of FGM interlayer thicknesses As in the preceding factor of safety case, similar behavior can be seen here That is, the specific factor of safety of the beam tends to behave no nevenly as the
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99 interlayer thickness increases. Again for low interlayer thicknesses from 0 to 0.00 05 m, the decreases ; for thicknesses from 0.00 05 to 0.00 50 m the in creases up to its maximum value of 0.253 However, differently from factor of safety case, f or thicknesses between 0.00 50 to 0.0062 m, the tends to flatten out its value to 0.25 3 As we can see, this value of the specific factor of safety is not that low compared to those found for low interlayer thicknesses. Finally, from Figure 51 and based in the specific factor of safety criterion, we can decide that the best FGM interlayer thickness for the given conditions is 0.0050 m where its has a maximum value of 0.253 Even though this FGM interlayer thickness (0.0050 m) seems to be relatively close to the one found using the criterion (0.0045 m ), for a different applications and/or conditions this small difference could be very significant especially in engineering applications that are highly sensitive to the geometrical parameters.
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100 Chapter 6 Conclusions and Future Work Introduction This chapter summarizes the findings of the analyses and the models studied in this dissertation Also, potential practical applications and benefits of this work within industry are discussed. Finally, recommendations for future research are made to supplement the modeling and analyzing techniques for functionally graded materials structures presented in th is work Conclusions From the s imulation results for the beam models, b oth elements (2 node and 3 node) perform equally in the example cases presented in terms of axial stress and transverse shear stress when calculated from the constitutive relations. However when the shearing stress is calculated using the equilibrium equation, only the three node elemen t perform s well. The inclusion of the axial gradient for the examples chosen does not alter the axial stresses significantly but does produce differences in the transverse shear stress as calculated from the equilibrium equation.
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101 The 3 node beam element mo del was implemented into a finite element code in MATLAB and code verification was performed on a composite cantilever beam. Benchmark comparisons of finite element predictions of stress field with the analytical solutions for a composite cantilever beam r esulted in a good agreement Simulations were also successfully performed on different beam models which demonstrate the ability of the 3 node beam element model to simulate thermo mechanical stresses in different structures and under different mechanical and thermal loading conditions. Comparisons of the element formulation with FGM models available in related literature are presented. In general, from the results of these comparisons, we can conclude that the qualitative and quantitative results obtained in this work are very similar to the referenced literature, which demonstrate s a suitable performance of the element formulations pres ented in this work. From the beam model s imulations with actual t emperature d istributions with and without t emperature d ependence of the thermo elastic material properties it was revealed that when temperature dependence is taken into account the tempera ture profile distributio n within the model is very different from the results obtained when temperature dependency is not considered. The heat insulation effect of alumina is higher when the temperature dependence is considered It was also found that intr oducing a FGM interlayer between the bi material beam model produce highe r heat insulation effect when comparing with the bi material and average interlayer models This fact can be use d in engineering applications where insulation effects need to be impro ved From the study of the influence of the FGM interlayer thicknesses on the f actor of s afety in beam structures constructed of FGMs under therm al loads it can be concluded
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102 that the inter layer thickness significantly influences the stress distribution factor of safety and the specific factor of safety of the structure In answer to the question posed in the introduction about how to implement element formulations for structures composed of FGMs it can be stated that the implementation involve d several steps: 1. the ability to integrate the variation of material properties through the thickness needs to be added to the mat erial library for beam elements ; 2. explore the effects of spatial temperature variation in the axial and through the thickness direction of the finite element ; 3. consider the influence of the temperature dependency of the material properties on the thermal stress es; 4. study the effect of the constituent layer thicknesses on the st resses factor of safety and specific factor of safety
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103 Recommendations and Future Work Even though the results found in this work were compared with the related literature, they should be used only as approximations, as further experimental testing should be used to verify the simulations results The following recommendations and future work is suggested 1. We recommend further investigation of f unctionally graded beam structures with material properties varying in directions other than through the thickness 2. One c ould develop a d esign of e xperiments study on the influence of the variables affecting the f actor of s afety/ mass ratio in structures constructed of functionally graded materials under thermo mechanical loads. This study w ould allow, among other benefits, an analysis/comparison of t he advantages/benefits of using structures constructed of functionally graded materials with respect to those constructed with homogenous materials 3. A further investigation regarding the techniques for estimating effective m aterial properties of functional ly graded materials is desirable In the graded layer of real FGMs, ceramic and metal particles of arbitrary shapes are mixed up in arbitrary dispersion structures. Hence, the prediction of the thermo elastic properties is not a simple problem, but complic ated due to the shape and orientation of particles, the dispersion structure and the volume fraction. This situation implies that the reliability of material property estimations becomes an important key for designing a FGM that meets the required perform ance
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104 References [1] Chakraborty, A., Gopalakrishnan, S., and Reddy, J. N., 2003, "A New Beam Finite Element for the Analysis of Functionally Graded Materials," International Journal of Mechanical Sciences, 45(3), pp. 519 539. [2] Suresh, S., and Mortensen, A., 1998, Fundamentals of Functionally Graded Materials : Processing and Thermomechanical Behaviour of Graded Metals and Metal Ceramic Composites IOM Communications Ltd, London. [3] Reddy, J. N., 1997, "On Locking Free Shear Deformable Beam Finite Elements," Computer Methods in Applied Mechanics and Engineering, 149(1 4), pp. 113 132. [4] Sankar, B. V., 2001, "An Elasticity Solution for Functionally Graded Beams," Composites Science and Technology, 61(5), pp. 689 696. [5] Red dy, J. N., 2004, Mechanics of Laminated Composite Plates and Shells : Theory and Analysis CRC Press, Boca Raton. [6] Na, K. S., and Kim, J. H., 2005, "Three Dimensional Thermomechanical Buckling of Functionally Graded Materials," A I AA Journal, 43(7), pp. 1605 1612. [7] Cooley, W. G., 2005, "Application of Functionally Graded Materials in Aircraft Structures," M.S. Thesis, Air Force Institute of Technology, Wright Patterson AFB OH. [8] Hill, M. R., and Lin, W. Y., 2002, "Residual Stress Measurement in a Cer amic Metallic Graded Material," Journal of Engineering Materials and Technology, 124(2), pp. 185 191. [9] Hill, M. R., Carpenter, R. D., Paulino, G. H., Munir, Z. A., and Gibeling, J. C., 2002, "Fracture Testing of a Layered Functionally Graded Material," Fracture Resistance Testing of Monolithic and Composite Brittle Materials, ASTM Special Technical Publication, 1409, pp. 169 186.
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About the Author Sim n A. Caraballo was born in Rio Caribe Sucre Venezuela in 196 6. He received the B.S. degree in Mechanical Engineering from UNEXPO University, Puerto Ordaz Venezuela, in 1991 and the M.S. degree in Mechanical Engineering from UNEXPO University, Puerto Ordaz Venezuela, in 19 96 He c ompleted the requirements for the Doctor of Philosophy degree in Mechanical Engineering at University of South Florida Tampa Florida in May 20 11 A summary of his professional and academic experience is as follow: Maintenance engineer at CVG Ferrominera Orinoco, Ciudad Piar, Venezuela, December, 199 1 December, 1994 Faculty of Department of Mechanical Engineering at UNEXPO University, Venezuela from December, 19 94 January, 2001. Research and teaching assistant in the College of Engineering at University of South Florida, Tampa, Florida, while pursuing the Ph.D. degree His research interests include finite element simulation, t hermo mechanical stresses in mechanical structures made of functionally graded materials nonhomogeneous and temperature dependent thermo elastic material properties He can be reached by e mail at s imon_ caraballo @ hotmail.com
